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# hw7 - 72 cHAprER z THE SCHRODNGER EQUATIoN rN oNE DIMENSIoN...

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Unformatted text preview: 72 cHAprER z. THE SCHRODNGER EQUATIoN rN oNE DIMENSIoN 7 . 1 1 o [ n e w ] ( a ) z z * : ( x + i y ) ( r - i A ) = 1 2 I y 2 = l z l 2 . (b) We first note that, as you can easily check, (zw)* : z*ru*. Then, lzwl2 = zw(2y1)" : zwz*w* : (zz-)(uw-) : lzl2lul2. Therefore lzwl: lzl.lul. (c) If v(r, t) : {(x)e-tut, then lv(r,t)l= l{(')t.le--tl : lr/(c)1. 7.L2 o [old,8.t0] (a) z =re-tut: r(cos &rt-i sin wt) = a-60, wherez: rcos(trt) andy:asi11r1;. 7.13 or [old 8.tt] (a) If. g : asst1r1l * bsin(c,.rl), we can define A: 1;T+F, so that ,4 is the hypotenuse of a triangle with sides a, b, and .4. If we call / the angle opposite a, lQ = arcstn(a/A)l then sin / : alA and, cos d : b/?4. T'hus a = Al(a/A) cos at + (blA)sinutl = -A[sin dcosut + cos /sint.rt] : Asin(tut + d). Conversely, if y -- Asin(ut ! d) : ,4 sin dcos t,,'t t Acos\$sinult, then we can define a: Asind and b: Acosd and then y : acosut * bsinut, (b) If g: Asin(ut + Q), we let t' =t+flu and then y: Asin(o-rt'). 7.L4 co [old,8.r2] eie : 7 t Uq + Uq'z Pt + (iq3 ftt + ... : t + i0 - 02 lzt - i03 lsl + 04 l4t + i02 lst -... : G - 02 /2t + 04 l4l -...) + i(0 - es /sl + 05 lil - ...) : cos I * isind. 7.15 oo [new] F]om Euler's relation, we know that ei(e+o) : cos(9 + {) + i sin(d + {). (I) Also, by the well known property of exponentials, ei(0+6):eieeid:(cosl+isint)(cosd+isin/):(cosgcos/-sindsin{)+i(sin0cos/+cosdsin/). Comparing this with Eq, (I) above and equating real and imaginary parts, we find that cos(0 + S) : cos , cos d - sin I sin / and sin(9 + /) : sin d cos d * cos d sin / 7.16 oo fold 8.13] FYom E,q. (7.121) we know that8....
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hw7 - 72 cHAprER z THE SCHRODNGER EQUATIoN rN oNE DIMENSIoN...

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